Everyone has given you great feedback, so I’ll just emphasize: your kids need you.
If you didn’t have children and wanted to lose a few more years waiting for this woman to get to the point where you finally walk away, that’s okay, you’re an adult. You can make your own choices.
But having this woman around your kids? Oh, no. It’s bad enough that they’re watching you put up with so much. I hope they’re young enough and there’s enough distance that they’re protected from the worst of it. But your ex is right to not be happy. Adults model life for kids…I know you don’t want them to grow up thinking addiction and lying and chaos is okay, right?
I would bet money she’s sleeping with her rehab guy, who sounds like someone you really don’t want in your life even second hand (or third, or a hundredth), let alone your kids’ lives. All the lies are just lies and if she comes home? You’re going to have this violent guy and his violent girlfriend and a ton of drama taking over your life.
Optimism can be a great thing. In this case, maybe you can be optimistic that you can find someone else who would actually appreciate you and be a good influence on your kids?
Wishing you strength and clarity.